On the Integer Domination Root Conjecture

Abstract

The domination integer root conjecture asserted that $0$ and $-2$ are the only integer roots of the domination polynomial $D(G, x)$ for any graph $G$. In this paper, we document a counterexample of order $n = 33$ possessing an integer domination root at $x = -4$. We provide the complete structural description of the graph $G_{33}$, present its exact domination polynomial $D(G_{33}, x)$, and demonstrate its exact rational factorization. Furthermore, we outline the structural gadget mechanism involving transfer matrices and $S$-unit branch cancellations that gives rise to non-trivial zero evaluation at $x = -4$.

Disclosure

“d verified exhaustively for small orders, we present a counterexample of order n = 33 that demonstrates x = −4 is an integer dom- ination root. The key mechanism, namely the S-Unit Branch Cancellation, was discovered with the assistance of generative AI tools. 2 The Counterexample G33 Consider the graph G33 = (V, E) of order n = 33 and |E| = 36 edges, having cyclomatic number m = 36 − 33 + 1 = 4. The vertex set is V = {0, 1, . . . , 32}. The structural core consists of a 4-cycle C4”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T19:15:26.226549Z

Structural counts